Minimum Stirrups and the Section-Size Limit (ACI 318-19 9.6.3)

The stirrup detailing limits a beam can fail even after passing the shear strength check.

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Example

You enter

Concrete strength f'c (psi) 4000
Stirrup yield fyt (psi) 60000
Web width bw (in) 12
Effective depth d (in) 21.5
Stirrup area Av, both legs (in²) 0.22
Factored shear Vu (kip, 0 = detailing only) 40
Lightweight factor lambda (1.0 normalweight) 1

You get

Minimum Av/s required 0.0100 in^2/in
Widest spacing meeting Av,min 22.0 in
Av,min triggered above Vu of 12.2 kip
Governing max spacing 10.75 in
Section ceiling phi (Vc + Vs,max) 122.4 kip

Details, formula, and sources

ACI 318-19 9.6.3.4 Av,min/s = greater of 0.75 sqrt(f'c) bw/fyt and 50 bw/fyt (the 50 bw/fyt floor governs below about 4,444 psi), the 9.6.3.1 trigger Vu > phi lambda sqrt(f'c) bw d that the 2019 edition put in place of the old 0.5 phi Vc rule, the 22.5.1.2 ceiling Vs <= 8 sqrt(f'c) bw d, and the 9.7.6.2.2 spacing cap of d/2 and 24 in, halved past Vs = 4 sqrt(f'c) bw d. A 12 x 21.5 in beam with #3 stirrups needs Av/s = 0.0100 in^2/in, so 22 in satisfies the minimum but the d/2 = 10.75 in cap governs; the section itself tops out at 122.4 kip no matter how much steel goes in. Table 9.6.3.1 exemptions not modeled. A design aid, not the engineer of record's stamped design.

Av,min/s = max(0.75 sqrt(f'c) bw/fyt, 50 bw/fyt); s(Av,min) = Av / (Av,min/s); trigger = phi lambda sqrt(f'c) bw d; Vs,max = 8 sqrt(f'c) bw d; phi Vn,max = phi (2 lambda sqrt(f'c) bw d + Vs,max); s <= min(d/2, 24 in), or min(d/4, 12 in) when Vs > 4 sqrt(f'c) bw d. (phi = 0.75)

The ACI 318-19 9.6.3.4 minimum shear reinforcement, the 9.6.3.1 Av,min trigger, the 22.5.1.2 Vs section-size ceiling, and the 9.7.6.2.2 stirrup-spacing maxima, by name.

ACI 318 is readable free through the ACI online reading room at concrete.org; 9.6.3, 9.7.6.2.2, and 22.5.1.2 are in the published code.

Estimate. AHJ and licensed professional govern.

Field names used by the API: fc_psi, fyt_psi, bw_in, d_in, av_in2, vu_kip, lambda, av_min_per_s, s_max_av_min_in, trigger_kip, s_max_in, phi_vn_max_kip

Minimum area Av,min/s is the greater of 0.75 sqrt(f'c) bw/fyt and 50 bw/fyt; the constant floor governs at ordinary strengthsACI 318-19 9.6.3.4
2019 trigger Av,min required where Vu > phi lambda sqrt(f'c) bw d, replacing the 318-14 0.5 phi Vc rule; Table 9.6.3.1 exemptions not modeledACI 318-19 9.6.3.1
Section ceiling Vs may not exceed 8 sqrt(f'c) bw d, so beyond phi (Vc + Vs,max) the section must growACI 318-19 22.5.1.2
Spacing maxima lesser of d/2 and 24 in, halved to d/4 and 12 in once Vs exceeds 4 sqrt(f'c) bw dACI 318-19 9.7.6.2.2